We are tasked with proving that given any sequence of integers $a_1, a_2, \ldots, a_n$, there exist integers $j$ and $k$ such that the sum $\sum_{i=j}^k a_i$ is a multiple of $n$.

Proof Outline:

1. Prefix Sums: Define the prefix sums of the sequence: $S_m = a_1 + a_2 + \cdots + a_m \quad \text{for } m = 1, 2, \dots, n.$ Additionally, define $S_0 = 0$ for convenience.

2. **R